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arXiv · 2606.13408

A catalog of fast matrix multiplication algorithms with exhaustive derivations

Benoit Chatain Lacelle
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Abstract

The 2022-2026 burst of activity in small-format matrix multiplication (AlphaTensor 2022, AlphaEvolve 2025, Schwartz-Zwecher 2025, and Perminov's open-source flip-graph framework) has produced striking results but scattered them across fields, attribution conventions, and serialisation formats. We present a unified, machine-checkable catalog covering shapes up to <32,32,32> over Q, Z, R, C, F2, and F3, with a separate axis for commutative algorithms (Waksman 1970, Makarov 1986, Rosowski 2019). Derivation over the catalog applies a fixed set of well-defined operators exhaustively - axis-flip, Kronecker (with its serendipitous "bud" variant), axis concatenation, recombination-with-allocation (with optional output peeling and leaf-level pair fusion), and downward projection - iterated to a fixed point. Our derivation layer is closest in spirit to Sedoglavic's FMM-Lille catalog, which likewise layers recursive derivations (Kronecker with serendipity, concatenation, projection) over known bases; we differ by covering more fields (Z, F2, F3, and a commutative axis alongside Q/R/C) and by exploring the derivation space more exhaustively - rather than sampling it stochastically, as the flip-graph / meta-flip-graph methods do. As of the 2026-07-12 snapshot over Q, our derivations improve on the best external catalog (FMM-Lille or Perminov) at 908 shapes - 19% of the 4760 comparable shapes in the large-format band 17 <= max(n,m,p) <= 32; in fact every strict win we produce lands in that band, small formats being saturated. We tie the external best at a further 3688 shapes and trail on only six audited ones, each explained by a single sharing device (fusion kernels) that our composition deliberately forgoes. We refresh the DIS09 comparison tables, split per field with a commutative column, and provide tooling to regenerate them as the catalog evolves.

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Benoit Chatain Lacelle. 2026-06-11. A catalog of fast matrix multiplication algorithms with exhaustive derivations. https://arxiv.org/abs/2606.13408

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